2 SPECTRUM OF AN ELEMENT. OF A NORMED ALGEBRA 315 that, if A s C — {0}, the series with general term u"(t)j\n+l is convergent for all / e E, and use the Banach-Steinhaus theorem (12.16.5).) 6. Let E be a separable Hilbert space and let (<?„)« 5=1 be a Hilbert basis of E. Let A be the Banach algebra &(E). (a) Let u e A be the endomorphism such that u(en) ~2"~nen+l. Show that u is quasi- nilpotent, but not nilpotent. (b) Define a sequence (an) as follows : if n is the product of 2k and an odd number, then an == e~k. Let u 6 A be the endomorphism defined by v(en) — ocnen + 1. Show that v is not quasi-nilpotent. (Observe that \\vk\\ = sup (am am+i • • • <xm+k^i), and evaluate m «i«2 •••a2r_r) (c) For each integer A:, let vk e A be the endomorphism defined as follows: vk(ej) = 0 if n is the product of 2* and an odd integer, vk(ea) — <x,nen+1 otherwise. Show that vk is nilpotent and that ||t? — vk\\ tends to 0 in A as A: tends to 4- oo. Hence the set of quasi-nilpotent elements of A is not closed. 7. Let A be a Banach algebra with unit element e. The radical of A is defined to be the set 9R of elements x e A such that p(ax) = 0 for all a e A. If x e 9R, then also p(xa) = 0 for all a e A, and 9ft is a closed two-sided ideal, equal to the set of all x such that e — ax is invertible for all a e A. (Observe that if u is invertible and x e 9*, then u — ax is also invertible for all a e A.) Every element of 9ft is quasi-nilpotent. If $ is a left ideal all of whose elements are quasi-nilpotent, then $ c 9ft. If 8ft = {0}, the algebra A is said to be without radical. In any case, A/SR is without radical. If TT : A -> A/9ft is the canonical homomorphism, then SpA00 = SPA/<R(TT(X)) for all x e A. 8. Let E be a Hilbert space and let A = -^(E) be the Banach algebra of continuous endomorphisms of E. (a) Let u e A. If u is not injective, then u is a left zero-divisor. If u(E) is not dense in E, then u is a right zero-divisor. If u is injective and u(E) is dense in E but not equal to E, then u is a left and right topological zero-divisor, but not a zero-divisor. (Remark that there exists a sequence (xn) of elements of E such that \\xn\\ = 1 for all w, and such that lim u(xa) = 0.) n-+oo (b) If u is surjective but not injective, or if u is injective and w(E) is closed in E but distinct from E, then u is an interior point of the set of non-invertible elements of A (use Problem 3(c).). (c) Show that A is without radical (show that if the radical were ^ {0} it would contain an orthogonal projection of rank 1). 9. Let A be a Banach algebra with unit element e. For each x e A the series x x2 x" is convergent; its sum is denoted by exp(x). If x, y are two elements of A which commute, then exp(# 4- y) = exp(x) exp(y). In particular, the image of C under the mapping f i—»exp(£jc) is a connected subgroup of the group G of invertible elements of A. For each x e A such that \\e — x\\ < 1, the series with general term — (e — x)"/n (n ^ 1) is convergent. If its sum is denoted by log x, then we have exp(log x) = x. Show that the subgroup of G generated by exp(A) is the connected component of the neutral element of G. Consider in particular the case where A is commutative.equence (a^mj]) tends to X(r)t